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A354797
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Triangle read by rows. T(n, k) = |Stirling1(n, k)| * Stirling2(n + k, n) = A132393(n, k) * A048993(n + k, n).
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2
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1, 0, 1, 0, 3, 7, 0, 12, 75, 90, 0, 60, 715, 2100, 1701, 0, 360, 7000, 36750, 69510, 42525, 0, 2520, 72884, 595350, 1940295, 2692305, 1323652, 0, 20160, 814968, 9549120, 47030445, 109794300, 120023904, 49329280, 0, 181440, 9801000, 156008160, 1076453763, 3723239520, 6733767040, 6065579520, 2141764053
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OFFSET
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0,5
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LINKS
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FORMULA
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Sum_{k=0..n} (-1)^(n - k)*T(n, k) = n^n. - Werner Schulte, Jun 03 2022 in A000312. [Formerly a conjecture, now proved by Mike Earnest, see link.]
T(n, k) = A132393(n, k) * A354977(n, k) = (1/n!) * Sum_{j=0..n} (-1)^(j + k) * binomial(n, j) * Stirling1(n, k) * j^(n + k).
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EXAMPLE
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Table T(n, k) begins:
[0] 1
[1] 0, 1
[2] 0, 3, 7
[3] 0, 12, 75, 90
[4] 0, 60, 715, 2100, 1701
[5] 0, 360, 7000, 36750, 69510, 42525
[6] 0, 2520, 72884, 595350, 1940295, 2692305, 1323652
[7] 0, 20160, 814968, 9549120, 47030445, 109794300, 120023904, 49329280
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MAPLE
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T := (n, k) -> abs(Stirling1(n, k))*Stirling2(n + k, n):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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